An essential step in describing a self-organized system is to quantify interactions between the system's components, which is commonly done within the framework of information theory. Using information theoretic measures, interactions can be characterized by order (pairwise, three-way, etc.), type (redundant, synergistic, independent, etc.), and magnitude. When applied to spatiotemporal systems, these measures act as an "information filter", revealing an underlying structure based on the interactions of the system's components. Previous work has demonstrated the value of such filters for identifying emergent patterns in spatiotemporal systems, primarily cellular automata (1). Here, we present a novel spatiotemporal filter based on the recently developed Partial Information Lattice (PIL) proposed by Williams & Beer (2). The primary advantage of our filter is that it separates interactions of all orders and types, thereby revealing nested levels of structure within a system. We demonstrate the utility of our approach by exploring interactions in elementary cellular automata (ECA) and a simple neural model.

Applied to ECA, the PIL can be used to filter interactions between a particular cell and its neighborhood by separating the information into three non-overlapping terms (synergy, redundancy, and unique information) of various orders. Our PIL filter shows a clear association between distinct information patterns and specific transient structures within ECA. For example, we find that the structures commonly referred to as "gliders", which have long been considered the mechanisms of information transfer in ECA, tend to be associated with high amounts of unique information. Similarly, when two gliders collide and result in a new product, we typically find a high amount of third-order synergy. This creation of a product is generally considered to be the mechanism of information modification in ECA. As we will show, other structures (such as the regular domain or "background") also have specific connections to information types of various orders. As a result, our filter can be used, for instance, to select for sites where information is being transferred without interference from sites of information modification.

The PIL filter can be applied to a variety of systems, which we demonstrate with a second example of a simple artificial neural network. We consider a neural model in which interactions between neurons are idealized as Boolean logic operations. We show that these logical interactions themselves can be quantified in much the same way as interactions between cells in ECA (3). Again, the advantage of the PIL is that it allows us to separate these interactions into non-overlapping terms. Similarly, different types of logical operations in a noisy neural model can be unambiguously determined using the information filters we have developed. We will briefly demonstrate this concept and compare our results with those based on existing information-theoretic metrics.

References (1) See "Local information transfer as a spatiotemporal filter for complex systems" by Lizier et al. (2008) for an example. (2) See "Nonnegative Decomposition of Multivariate Information" (2010) by Williams & Beer for more information. (3) See "Network Information and Connected Correlations" by Schneidman et al. (2003) for an example.